The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution ((P) over bar(t)) of this Riccati equation. This solution is attracting, in the sense that if P-t is another solution, then P-t - (P) over bar(t) converges to 0 exponentially fast as t tends to +infinity, at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices.